By A. Knyazev (speaker) and K. Neymeyr.
Abstract: We first review ugly, but powerful, convergence rate bounds of [1] for symmetric generalized eigenvalue problems. Then a new elegant approach, is presented, see [2], that gives a geometric proof of a sharp convergence rate bound of a simple preconditioned eigensolver--the gradient iterative method with a fixed step size, where we use the gradient of the Rayleigh quotient as an optimization direction. The crucial step of the new proof is a reduction of the convergence analysis to a two-dimensional subspace spanned by relevant eigenvectors, based on analyzing the gradient flow of the Rayleigh quotient. However, it is not currently known if this approach of [2] can be directly extended to the subspace iterations to improve and simplify the results of [1], which thus remain unbeatable. [1] J. Bramble, J. Pasciak, and A. Knyazev, A subspace preconditioning algorithm for eigenvector/eigenvalue computation, Advances in Computational Mathematics, 6 (1996), no. 2, 159--189. [2] A. Knyazev and K. Neymeyr, Gradient flow approach to geometric convergence analysis of preconditioned eigensolvers